Problem: Simplify; express your answer in exponential form. Assume $k\neq 0, a\neq 0$. $\dfrac{{(ka^{-1})^{-1}}}{{(k^{2}a^{4})^{5}}}$
Answer: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(ka^{-1})^{-1} = (k)^{-1}(a^{-1})^{-1}}$ On the left, we have ${k}$ to the exponent ${-1}$ . Now ${1 \times -1 = -1}$ , so ${(k)^{-1} = k^{-1}}$ Apply the ideas above to simplify the equation. $\dfrac{{(ka^{-1})^{-1}}}{{(k^{2}a^{4})^{5}}} = \dfrac{{k^{-1}a}}{{k^{10}a^{20}}}$ Break up the equation by variable and simplify. $\dfrac{{k^{-1}a}}{{k^{10}a^{20}}} = \dfrac{{k^{-1}}}{{k^{10}}} \cdot \dfrac{{a}}{{a^{20}}} = k^{{-1} - {10}} \cdot a^{{1} - {20}} = k^{-11}a^{-19}$